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Ideology and existence of 50% : Majority equilibria in multidimensional spatial voting models
Hervé Crès, Utku Unver
To cite this version:
Hervé Crès, Utku Unver. Ideology and existence of 50% : Majority equilibria in multidimensional spatial voting models. 2005. �hal-01065582�
Ideology and existence of 50%-majority equilibria in multidimensional spatial voting models ∗
Hervé Crès HEC Paris
†M. Utku Ünver Koç University
‡21 June 2005
Abstract
When aggregating individual preferences through the majority rule in an n-dimensional spatial voting model, the ‘worst-case’ scenario is a social choice conÞguration where no political equilibrium exists unless a super majority rate as high as 1−1/n is adopted. In this paper we assume that a lower d-dimensional (d < n) linear map spans the possible candidates’ platforms. Thesed ‘ideological’ dimensions imply some linkages between the n political issues. We randomize over these linkages and show that there almost surely exists a 50%-majority equilibria in the above worst-case scenario, whenngrows to inÞnity.
Moreover the equilibrium is the mean voter. The speed of convergence (toward 50%) of the super majority rate guaranteeing existence of equilibrium is computed for d = 1and 2.
Keywords: Spatial voting, super majority, ideology, mean voter theorem, random point set.
∗This research started while H. Crès was visiting Koç University. H. Crès would like to thank the CASE for its hospitality and the Fondation HEC for Þnancial support. M. U. Ünver gratefully acknowledges support from the NSF and TÜBA-GEBúIP (Turkish Academy of Sciences - Distinguished Young Scholar Program).
†E-mail: cres@hec.fr; address: HEC School of Management, 78351 Jouy-en Josas, France; tel: (+33) 1 39 67 72 36; fax: (+33) 1 39 67 70 85. Hervé Crès is a member of the GREGHEC, unité CNRS, FRE 28-10.
‡E-mail: uunver@ku.edu.tr; web-site: http://home.ku.edu.tr/~uunver; address: Koç University, Department of Economics, College of Administrative Sciences and Economics (CASE), Rumelifeneri Yolu, Sarõyer,Istanbul, 34450, Turkey; tel: (+90)-212-3381354; fax: (+90)-212-3381642.ú
1 Introduction
It is well known since Plott (1967) that a 50%-majority stable political equilibrium typ- ically does not exist in a multidimensional voting setup. A way to restore existence of a stable outcome is to require a super majority rule to overrun the status quo, thus giving rise to the concept of ρ-majority equilibrium, where ρ∈ [1/2,1] is the proportion of the voting population a challenger must rally to take over. It is widely admitted that the smaller the rate of super majority needed to secure existence of an equilibrium (i.e., the less conservative the voting rule), the better.
There is a wide literature on the level of super majority required for existence, both in deterministic or probabilistic setup (see, e.g., Ferejohn and Grether (1974), Caplin and Nalebuff (1988, 1991) and Balasko and Crès (1997)). In a standard social choice setup where agents, endowed with continuous and convex preferences, have to choose among political alternatives in a non-empty, compact and convex subset ofRn, Greenberg (1979) shows that a necessary and sufficient condition for the existence of a ρ-majority equilibrium is ρ≥ n
n+1.
To show that this bound is tight, Greenberg (1979) constructs a voting conÞguration where no incumbent is stable with respect to a super majority rule with rate smaller than n+1n . It follows: Take n+ 1 independent points in Rn and interpret them as the ideal political choices of n+ 1voters endowed with euclidean preferences. Denote Sn the n-dimensional simplex generated by the voters’ ideal points. Fix an incumbent x /∈ Sn; then s(x) = argmin {kx−sk, s ∈ Sn} is unanimously preferred to x, hence x is not stable under any ρ-majority rule with ρ < 1. Now, Þx an incumbent x ∈ Sn; then it is always possible to Þnd a challenger preferred by n out of the n+ 1 voters: indeed, denoteS¯n the(n−1)-dimensional simplex generated by the ideal points of thesenvoters (S¯n is a face of Sn), then one can reconduct the previous argument, and show that
¯
s(x) = argmin{kx−sk,¯ s¯∈S¯n}is preferred to x by all of these nvoters.
This example is thus a ‘worst-case’ scenario. One easily sees that if the voters’ ideal points are taken in a lower dimensional subspace, then the upper bound decreases. But the gain remains small though. And one gets existence of political equilibria for not too conservative voting rules only when the number, n, of political issues is very low. This bound isρ= 1/2whenn= 1 (the so-called ‘median voter theorem’);ρ = 2/3whenn= 2;
and forn≥3, then the required rate of super majority must be above 3/4 (and converges the the unanimity criterion when n goes to inÞnity), a level very rarely observed in practice. Indeed, constitutions or corporate charters build on super majority rates which are very rarely above 70%1, although the number of political issues at stake in electoral
1For decisions on issues which are delegated to the European Union, the rate was 72% in the Maastrich Treaty, it was decreased to a mix of 65% —of the States— and 55% —of the population— in the Constitutional
processes is obviously often very large: it is not rare, when reading political platforms proposed by candidates in large elections, to denumerate several dozens of issues2. Hence the question: why, if there are so many issues, do we observe so reasonable super majority rates in practice?
AÞrst answer might be that one should not believe in Greenberg’s worst-case scenario.
A second answer can be found in the Hinich-Ordeshook spatial voting model3. According to the latter, there are only a few political dimensions underlying the platforms proposed by the candidates. These few dimensions are claimed to be ideological. Ideologies imply linkages4 between political issues and thus span a lower dimensional linear space (dubbed the ‘campaign space’ in the sequel) on which the original distribution of voters’ ideal points is projected.
The assumption that political platforms are based on ideology stems from the belief that the cleavages between candidates separate along simpler, more predictable lines than then-dimensional policy space would imply. As Popkin (1994) states it (p. 51): “Ideology is not the mark of sophistication and education, but of uncertainty and lack of ability to connect policies with beneÞts... Parties use ideologies to highlight critical differences between themselves, and to remind voters of their past successes”. This approach has some empirical relevance: Poole and Rosenthal (1991, 1996) show that in the USA, with the exception of the 32nd Congress, two dimensions are always capable of explaining more than 80% and up to 95% of the variation in the votes of elected officials on most issues. The same, Poole and Rosenthal (1997) and McCarty, Poole and Rosenthal (1997) test the Hinich-Ordeshook spatial voting model on post World War II Congressional roll call voting and show that only two dimensions are required to account for most of the votes: the liberal-conservative continuum5 and the dimension of conßict over race and civil rights.
But one cannot exclude that the number of underlying ideological dimensions be larger than 3. In the political debate in France on the referendum for ratiÞcation of the European
Treaty.
2Everybody in France recalls the ‘110 propositions’ of the candidate François Mitterrand for the presidential election of 1981.
3This model was Þrst proposed by Cahoon, Hinich and Ordeshook (1976), Ordeshook (1976) and Hinich and Pollard (1981) and then developed by Enelow and Hinich (1984) and Hinich and Munger (1994).
4These linkages formalize the fundamental insight of Converse (1964) according to which ideology (the Conversian ‘belief system’) interrelates and bundles the political issues: ideology is fundamentally the knowledge of what-goes-whith-what. As Converse (1964) states it: ideology is “...a conÞguration of ideas and attitudes in which the elements are bound together by some form of constraint” (p. 207).
5A judgemental dimension that has been “highly serviceable for simplifying and organizing events in most Western politics for the past century”, Converse (1964, p. 214).
constitutional treaty during the Spring of 2005, one cannot explain the cleavages between and within parties through the traditional left-right dimension. One also needs the now classical ideological dimension ‘sovereignist-federalist’ to explain the split of the gaullist party; furthermore the possible future entry of Turkey was an element of the debate, and religion (the ideological position in the ‘laïcity’ dimension) was clearly the only way to explain another (orthogonal) split of the gaullist party.
Another type of political debates often builds on more than three underlying dimen- sions: proxy Þghts in publicly traded corporations in a context of market failures6. The stakes are probably simpler to grasp than in ordinary political debates; moreover, share- holders usually have access to a more measurable and precise information which is easier to aggregate. Yet corporate charters rarely choose rates of super majority beyond 65%.
The answer to the question “why do we observe so reasonable super majority rate in practice?” seems to be: not only the political competition articulates along fewer, simpler and more predictable lines than the n-dimensional policy space would imply, but also one should not believe in Greenberg’s worst-case scenario. The present paper goes one step further. Its main contribution is an aggregation theorem that links the two latter arguments: one should not believe in Greenberg’s worst-case scenario because the political competition happens in a lower dimensional subspace spanned by the underlying ideologies. Indeed, if we randomize on the linkages between issues imputed by ideologies, our main result (Theorem 1) states that the Hinich-Ordeshook approach almost surely transforms Greenberg’s worst-case scenario into the best-case scenario of a symmetric distribution of voting characteristics. And as a consequence we obtain (Theorem 2) a mean-voter theorem: the mean voter happens to almost always be the unique 50%- majority equilibrium, when the number of political issues grows large.
The paper is organized as follows: Section 2 introduces the model, Þrst the classical Downsian spatial voting model (Section 2.1), then its Hinich-Ordeshook sophistication (Section 2.2). Then Section 3 states and proves the aggregation theorems. Section 4 computes a lower bound to the speed of convergence of the expected min-max rate toward 50%; the computations give upper bounds on the expected rate of super majority necessary to sustain the mean voter as a political equilibrium, for any number of voters, when the number of underlying ideologies is smaller than 2. Section 5 ends the paper with some concluding comments.
6The heterogeneity of the shareholders’ opinions can come from imperfect competition, the incomplete- ness of Þnancial market structure or the presence of externalities. ‘Ideological’ dimensions in corporate politics can be: the ‘philosophy’ with respect to debt vs equity, horizontal vs vertical integration, inter- national diversiÞcation, expansion vs concentration...
2 The model
2.1 Voters, platforms and the majority rule
The setup to model the electoral process and voting mechanism is the classical Downsian multidimensional spatial voting model (Downs (1957)). There are n measurable criteria of political activity, so that a political platformin the policy space can be represented as ann-dimensional vector: x∈Rn. There aremvoters in a setI. Each voter is endowed with an euclidean preference relation on Rn: agent i, 1 ≤ i ≤ m, has a preferred choicein the policy space,xi ∈Rn, and his/her utility function over the space of political choices is decreasing with the euclidean distance from his/her preferred choice:
∀x∈Rn ui(x) =−kxi−xk Asociety is am-tuple X = (xi)mi=1.
We measure the stability of a political platform in a given society through the Simpson- Kramer approach. Given two political choices(a, b)∈Rn×Rn, ρ(b, a)measures the ratio of the electorate that strictly prefers btoa:
ρ(b, a) = /{i∈I|ui(b)> ui(a)}
m .
Thescoreof a political choicea ∈Rn is: ρ(a) = maxb∈Rnρ(b, a). Clearly, the score of any political choice taken outside the closed convex hull, hXi, of X will be 1: the challenger b that minimizes the distance between a and hXi is unanimously preferred to a. Hence looking for the ‘best’ status quo, i.e., the ones with lowest score, we can reduce our search to hXi. The min-max rate of society X is: ρ∗ = mina∈Rnρ(a). The min-max set of societyX is: S∗(X) ={a∈Rn|ρ(a) =ρ∗}.
Themajority rule with rateρ∈[0,1]states that candidatebis preferred by society X to (or defeats) candidate a if and only if ρ(b, a) > ρ. A candidate a is said to be ρ-majority stable in societyX if and only if there is no alternative that defeats it, i.e., if and only if its score is not larger than ρ: ρ(a) ≤ ρ. Such a candidate is a political equilibrium for the majority rule with rate ρ.
One knows since the seminal work of Plott (1967) that 50%-majority stable equilibria generally do not exist when n ≥ 2. To recover existence of political equilibria, one has to impose a super majority voting rule, i.e., a voting rule with rate ρ >1/2. This paper deals with existence of such political equilibrium based on super majority voting. Along that search, political platforms in the min-max set have this appealing property that they are equilibria for the lowest rate of super majority, hence theless conservative voting rule.
The super majority rate one has to impose in order to recover existence of equilibrium can be quite high, though: As extensively explained in the introduction, suppose that
m =n+1and them-tupleX are the vertices of ann-dimensional simplex, then obviously any political choice in hXi has a score of n/(n+ 1) (it is enough to choose a challenger closer to any of then+ 1 (n−1)-dimensional faces of the simplex). Therefore, since any political choice outside hXihas score 1, the min-max rate is n/(n+ 1) and the min-max set is hXi: one has to impose a super-majority rule of rate ρ≥n/(n+ 1) to get a stable political choice, and then all choices in hXi are ρ-majority stable. Greenberg (1979) proves that the condition ρ≥n/(n+ 1)–to get existence of aρ-majority stable political equilibrium– is in fact sufficient as soon as the voter’s preferences satisfy very mild properties of continuity and convexity. Hence, the latter case is aworst-case scenario, as far as getting not too conservative min-max rate is concerned. The present paper can be read as an attempt to downside the relevance of this worst-case scenario.
Some convincing arguments along the same line are available in the social choice literature. One of the most important one is given in Caplin and Nalebuff (1988, 1991).
They give a dimension-free upper bound to the min-max rate under the conditions that preferred choices of agents are selected from a σ-concave distribution with compact and convex support. This upper bound (which, asymptotically, is lower than 64%) is given by the score of the mean voter, the voter whose preferred choice is the barycenter of all xi’s.
This literature can roughly be regarded as looking for multi dimensional versions of the median voter theorem.
2.2 Ideology, candidates and political campaigns
A central assumption of our model is that, although the number (here: n) of criteria for political activity can indeed be quite large, the political competition takes place in a subspace of lower dimension: d < n. In accordance with the Hinich-Ordeshook spatial voting model, this lower dimensional space is considered to be the ideological space, assumed to be Rd without loss of generality. According to this approach, the ideologies are linked to the platforms by a linear map, L, from the ideological space to the policy space: a candidate, πA ∈ Rd, imputes a platform xA ∈ Rn such that xA = x0 +LπA, wherex0 is the platform of status quo policies. Finally, thed-dimensional affine subspace which is the image of Rd by L translated byx0 is called the campaign space, C ⊂Rn, in the sequel. Before developing the strength of the model, let us illustrate through an example how the linear map Loperates.
An illustration: Issues of political activity are often precise and technical; consider two such classical issues like (1) how much of the State’s budget, x1, must be allocated to buy helicopters, and (2) how much of the State’s budget, x2, must be allocated to create more slots in kindergartens. For the sake of simplicity, we limit the issues to these two, hencen= 2. The assumption is made that platforms proposed by candidates in this two-
dimensional policy space can be explained through a (say) one-dimensional underlying linear subspace, e.g., the classical liberal-conservative (left-right) dimension; henced= 1.
Given a vector of status quo policies x0, the sensitivity of xj, j = 1,2, to the position π of the candidate in the ideological space is a Þxed scalar lj ∈R, therefore xj is anaffine function of π:
xj =x0j +ljπ , j = 1,2 .
Figure 1 (resp. 2) plots the relation between ideology and helicopters (resp. kinder- gartens). E.g., the policy regarding kindergarten is almost not sensitive to ideology, and only slightly decreases (l2is small and negative) withπ: a leftist candidate wants to create slots in kindergartens because these structures are more used by low-class workers than by wealthy families; a rightist candidate uses slots’ creation in kindergartens as an incentive to increase fertility. The policy regarding helicopter is more sensitive to ideology, and increasing (l1 is positive): rightist candidate are usually more hawkish, and spending on helicopters rises as ideology moves right, as shown by the plain line L1.
Figure 1 Figure 2
L1
L01
π π
x1 x2
x01 x02
- -
6 6
The sensitivity of policies to ideology as depicted on Figures 1 and 2 implies a linkage between the two issues in the policy space: the induced campaign space, C (plain line on Figure 3), is going through the status quo x0 with slope l2/l1. The induced ‘ideal candidate’ x¯i of voter i, whose preferred platform is xi, obtains by orthogonal projection of xi on the campaign space. And consequently, voter i votes for the candidate whose imputed platform is closest to his ‘ideal candidate’ x¯i. In the general (n, d) case, the euclidean structure of the original voting conÞguration gives rise, through the orthogonal projection onC, to a social choice problem involvingm voters with euclidean preferences in Rd. Hence we are dealing with a d-dimensional spatial voting problem with m voters and thus we are left with a combinatorial problem about m-tuples of points in Rd rather than in Rn.
The assumption that political platforms are based on ideology stems from the belief
Figure 3
x1
x2
x01 x02
x0•
C C0
¯ xi
¯ x0i
• xi
•
•
- 6
that the cleavages between candidates separate along simpler, more predictable lines than the n-dimensional policy space would imply. Simplicity and predictability makes the voter’s duty of voting easier. Not only ideology transmits information to voters, but also it creates enthusiasm for political action. This is a virtue of this approach on the candidate’s side. Of course, reducing the dimensionality of the campaign space also makes communication easier and less costly for candidates. But on top of that, the credibility of its commitment is stronger when his/her actions are perceived by the voters to be based on ideology7 (see Enelow and Hinich (1984) or Hinich and Munger (1997) for development of these arguments). Last but not least, as underlined in the introduction, the empirical relevance of this approach has been underlined by Poole and Rosenthal (1991, 1996, 1997) and McCarty, Poole and Rosenthal (1997).
If the Hinich-Ordeshook spatial voting model has the virtue of offering a more realistic view of electoral competition, we argue in the present paper that it moreover hasextremely nice properties as far as aggregation of individual preferences is concerned. Indeed, we prove in the sequel that the worst-case conÞguration of the society (worst case as far as aggregation is concerned) —i.e., when the point-setXis an(n+1)-tuple of points forming a n-dimensional simplex with equal voting rights on the vertices— transformsalmost surely
7As Hinich and Munger puts it, “classical spatial theory assumes that each politician chooses the position that maximizes his or her vote share given the expected platform of the opponents. (...) Such an approach (...) may be of little use in describing real world politics. For such an approach to work, voters must believe that a candidate who takes a position is likely to deliver that position. (...) The candidate’s promise must be ‘credible’.
into a best-case conÞguration —i.e., the (n+ 1)-tuples of projected points in Rd X is symmetrically distributed.
The Þrst step of our argument is to qualify what we mean by ‘almost surely’. Let us go back to the above illustration. Suppose now that an exogenous historical shock occurs, e.g., a terrorist attack. Most probably this event is going to impact the sensitivity of the Þrst issue (helicopters) to ideology: all candidates become hawkish and want to invest more into such a modern defense tool as helicopters, independently of his/her ideology.
Hence a new line L01, with a much smaller sensitivity rate: l01 < l1 (see the –almost ßat– dotted line on Figure 1). It is probably going to be the case that everybody in the society, candidates and voters are going to prefer an absolute increase ∆x1 > 0 in the political platform; we assume that in such an event the perturbed status quo becomes:
x010 = x01 +∆x1, and that for all i, the preferred platform’s Þrst component becomes:
x0i1 =xi1+∆x1. Hence this general absolute increase∆x1 results in aglobal (rightward) translation of the spatial point-set conÞguration, and this translation has no impact on the geometric properties of our problem. Therefore, without loss of generality, we can consider ∆x1 to be zero, and the only impact of this exogenous historical event is a drop in the sensitivity ratel1. This results into a new campaign spaceC0 (dotted on Figure 3) going through the status quox0 with slope l2/l10. The new induced ‘ideal candidate’ x¯0i of voter i obtains by orthogonal projection of xi on the new campaign space C0.
The idea is that such random shocks always happen, although fortunately not all as dramatic as a terrorist attack, and that their ‘media’ treatment and destiny can change the sensitivities of various issues to ideology. Then the central question is: How is this d-dimensional campaign space chosen? In the present paper, we take a purely Laplacian perspective and assume that C is selected at random, according to a ‘uniform’ distrib- ution on the natural underlying space. We deÞne C as an element in the Grassmanian G(n, d) of orientedd-subspaces inRn. Random historical and mediatic shocks generate a probability distribution over G(n, d). Among the latter ones, one arises ‘naturally’: the unique rotation-invariant probability measure, µ(n, d) (known as the Haar probability measure), on G(n, d), which intuitively selects all d-dimensional campaign spaces ‘with equal probability’. Hence µ(n, d) will be dubbed impartial in the sequel. The idea be- hind impartiality is that the main themes at stake in a political campaign depend heavily on the exogenous shocks of recent history, and the exogenous treatment by the media of these shocks.
3 Main result
For any selected campaign spaceC, the original social choice problem characterized by the point setX inRn gives rise to a lower dimensional social choice problem characterized by the (orthogonally projected) point setXinRd. SupposeXis am−1dimensional simplex inRn, such that for eachi,xi (a column vector inRn) is theith vertex of simplexX. We say that X is regular ifkxi−xjk=kxi−xkkfor any i, j, k.The simplex isO-centered if P
ixi = 0. Note that for anO-centered regular simplex, we havexTi xj =xTi xk<0for any i, j, k. In this section, we mute the translational part of the political shock as explained in the previous section and focus on the rotational shock to the campaign space. First, we will consider rotations pivoted at the center of gravity of the simplexX, that is, when the mean-voter is the status-quo (i.e. it is always on the campaign space). Center of gravity is normalized to O, center of the coordinate system. As we will show, this has no cost in our approach, since the center of gravity is the unique 50%-political equilibrium (Theorem 2).
Theorem 1 Let the campaign spaceC be impartially randomly selected. The point setX coincides in distribution with a negatively correlated sample from a symmetric probability distribution in Rd which becomes asymptotically independent as n → ∞ with the rate n1 when X is a regular O-centered simplex.
Proof of Theorem 1: Let n ≥ m > d. Take a regular (m−1)-dimensional O-centered simplex inRn whose vertices are column vectors of ann×m matrix X = [x1 x2 ... xm].8 We have xTi xj < 0 for any i and j. Let Π be d×n with πii = 1 for i ∈ {1,2, ..., n}
and all other entries of Π are 0. We can denote the random d−subspace by C=ΠR where R is a random rotation matrix distributed with the Haar probability measure among the n×n rotation matrix group denoted by R(n), that is, every n×n rotation matrix is chosen with equal probability as a draw of R. Note that C is distributed with probability measure µ(n, d) in GrassmanianG(n, d). Note that every rotation matrix is an orthogonal matrix. LetO(n)denote then×northogonal matrix group. First, we will consider orthogonal matrices instead of rotation matrices. Let C∗ =ΠA be such thatA is a random orthogonal matrix distributed with the Haar probability measure in O(n), that is, every n×n orthogonal matrix is chosen with the same probability as a draw of A. Note that orthogonal transformations include rotoinversions (where det (A) = −1) and rotations (det (A) = 1), we will rule out rotoinversions later. First note that every column of Ahas a symmetric distribution andEA= 0, since if Ais orthogonal then−A is orthogonal and both A=A and A =−A are equally likely events. Since orthogonal
8Let(i, j)th entry of a matrixH be denoted byhij andjth column vector ofH be denoted byhj.
transformation preserves the inner-product of two vectors, we have P
iaijaik = 0 for j 6= k and P
ia2ij = 1 for any orthogonal matrix A. Since every row permutation and column permutation of Ais equally likely to occur, E¡
a2ij¢
andE(aijak+)is constant for everyi, j, k and?. First, P
iE¡ a2ij¢
= 1impliesE¡ a2ij¢
= n1. Moreover,P
iE(aijaik) = 0 impliesE(aijaik) = 0, implying with the symmetry argument thatE(aijak+) = 0for every i, j, k, ? such that i 6=k or j 6=?. We are interested in the distribution of the columns of P =C∗X = ΠAX, the projection of X to the random subspace C∗. We will show that each column vector has identical symmetric distribution and each pair of column vectors are negatively correlated. The two eventsP=ΠAX andP=Π(−A)X are equally like to occur, therefore each column ofPhas a symmetric distribution. Since eachxihas the same length, each column vector Axi is an identical random vector. Let Σij =cov(Axi,Axj) for i6=j. The (g, h)th entry of Σij is σijgh=E((P
kagkxki) (P
+ah+x+j)). Forg 6=h, σijgh=X
k
X
+
E(agkah+)
| {z }
=0
xkix+j = 0
For g =h, σijhh=X
k
X
+
E(ahkah+)xkix+j =X
k
E¡ a2hk¢
| {z }
=n1
xkixkj+X
k
X
+6=k
E(ahkah+)
| {z }
=0
xkix+j = 1
nxTixj <0
Since pi is the Þrst d coordinates of Axi for each i, different coordinates of each pair of pi and pj are independently sampled and the same coordinates of each pair pi and pj are negatively correlated, where correlation goes to zero as n → ∞ (or m → ∞ since m ≤ n). We conclude our proof by observing that by Theorem 1 of Baryshnikov and Vitale (1994), above matrix A can be swapped with a random rotation matrix R, and
point setX consists of draws of columns of P=CX. ¥
Our next result states that there is a unique political equilibrium at the mean voter for 50%-majority rule as the number of voters goes to inÞnity.
Theorem 2 (Mean Voter Theorem) Fix d. Take m → ∞ then almost surely O, the mean voter, is the unique political equilibrium for the 50%-majority rule.
Proof of Theorem 2: Let f : Rd → R+ be the underlying limiting marginal probability density function (p.d.f.) for the columns of limm→∞X. Let Eρ(w) denote the expected value of the score of any point w∈Rn.
First, we show that O(∈Rn), the mean voter, is a political equilibrium for the 50%
majority rule. Negatively correlated sampling of political positions has a smaller min-max rate than independent sampling of political positions from the same symmetric marginal
distribution. Let Y be a set of points independently sampled from p.d.f. f. Under independent sampling, by Theorem 3 in Caplin and Nalebuff (1988) the min-max rate of O for Y converges almost surely to the min-max rate of f at O ∈ Rd when m → ∞. The min-max rate of Oforf is 0.5, sincef is symmetric aroundO¡
∈Rd¢
by Theorem 1.
Since min-max rate cannot smaller than 0.5, min-max rate of Ofor X also almost surely converges to 0.5, that is,limm→∞Eρ(O) = 0.5,implying O(∈Rn), the mean voter, is an equilibrium point for the 50%-majority rule.
Next, we prove that O(∈Rn) is the unique equilibrium issue. Take any inÞnite di- mensional issue vectorw6=O. ForÞniten, let theÞrstnentries ofwbe relevant. We will show thatlimm→∞Eρ(w)>0.5. LetC=ΠR be the random campaign space spanned by the Haar measure, like in the proof of Theorem 1 where Ris then×nimpartial random rotation matrix distributed with Haar measure on R(n) and Π is the d×n matrix with πii = 1 for i≤ d and all other entries of Π are zero. The random projection of the issue vector w on the campaign space is ΠRw. Note that ΠRw = O ∈ Rd with probability 0 and ΠRw 6= O ∈ Rd with probability 1, since w 6= O ∈ Rn. Hence, the only relevant draws of Rfor the calculation of Eρ(w)are allR∈R(n)such thatΠRw 6=O∈Rd. Fix a rotation matrix R ∈ R(n) such that ΠRw 6= O. Since f is symmetric around O ∈ Rd by Theorem 1, the min-max rate of the point ΠRw for f is greater than 0.5. Hence ex- pected min-max rate of the projection ΠRw is greater than 0.5 when m → ∞, implying limm→∞Eρ(w)>0.5and concluding thatwcannot be stable under 50% majority voting
rule. ¥
This result is inspired from a modern approach (in the literature from discrete and computational geometry) to generate random points. When dealing with ad-dimensional spatial voting problem with m voters, one is left with a combinatorial problem about m-tuples of (random) points in Rd. Many ‘natural’ distributions of these random points have been proposed in the mathematical literature (see Schneider (2004)). Among them, the one described above takes a central place: Every conÞguration of m > d numbered points in general position in Rd is affinely equivalent to the orthogonal projection of the set of numbered vertices of a Þxed regular (m−1)-dimensional simplex onto a unique d- dimensional linear subspace inRn. This construction builds a one-to-one correspondence between the (orientation-preserving) affine equivalence classes of such point set conÞgu- rations and an open dense subset of the Grassmanian G(n, d)of oriented d-spaces inRn. The so-called Grassmann approach (sometimes referred as the Goodman-Pollack model) considers the probability distribution on the set of affine equivalence classes ofm-tuples in general position inRd that stems from the unique rotation-invariant probability measure onG(n, d). Baryshnikov and Vitale (1994) (following an observation of Affentranger and Schneider (1992)) proved that under the Grassmann approach, the resulting point set
coincides in distribution with a standard Gaussian sample in that subspace. As a conse- quence, an affine-invariant functional of m-tuples with this distribution is stochastically equivalent to the same functional taken at an i.i.d. m-tuple of standard normal points in Rd.
Our model can be seen as giving another interpretation to the Grassmann approach:
We do not use it as a random generation of social choice conÞgurations (m-tuples of points in Rd) according to a ‘natural’ probability distribution, but as a random generation of a lower dimensional campaign spaces for any original (higher dimensional) social choice conÞguration. The latter in particular can be a worst-case scenario as deÞned in the introduction and Section 2.
Remark: A direct consequence of the Grassmann approach is that if we depart from a (regular) worst-case scenario in Rn and take X to be the vertices of a regular (n−1)- dimensional simplex (which can as well be translated to be spherico-regular), then, up to an affine transformation, the resulting point set X coincides in distribution with a standard Gaussian sample in Rd. In our approach, we do not need linear images and affine translations, since economically they do not correspond to any interpretations, and mathematically we do not require stochastic independence and an underlying Gaussian distribution.
4 Low dimensional social choice con Þ gurations
Of course, the asymptotic result in Theorem 2 does not give much idea of the rate of super majority, ρ(O), that one should impose in order to have existence of a political equilibrium when d andm are ‘small’, nor does it give an idea of itsexpectation, Eρ(O).
In this section, we give anupper-bound todistribution-free value of this expected score for d= 1 andd= 2. We considerindependent sampling of mpoints drawn from a symmetric distribution about O (∈ Rd for d = 1 and d = 2) which approximate the sampling for n→ ∞. In our approach in Theorem 1, we neednegatively-correlated sampling for Þnite n. Therefore, the results derived in this section are upper bounds to actual Eρ(O) for Þnite n when d = 1 and d = 2 where X is regular and O-centered. Let n → ∞ in this section. For uni- and bidimensional campaigns, the upper-bounds that we derive are plotted at the end of the section in Figure 7.
Unidimensional campaigns
Proposition 1 Set d = 1. Consider an m-sample independently drawn from a distribu- tion which is symmetric about O, then the expected value of the score of O is
Eρ(O) = 1
2 + 1 22[m2]+1
à 2£m
2
¤
£m
2
¤
!
where [x] denotes the integer part of x.
Proof of Proposition 1: Consider the following process which is due to Wendel (1962):
choosemrandom points in an interval centered atO: q1,q2,. . .,qm. For eachi,1≤i≤m, set pi equal to qi or to −qi with equal probability 1/2. The points p1, . . ., pm are again i.i.d. random points in the interval (we ignore the degenerate conÞgurations, occurring with probability zero, where two points qi and qj would be equal or opposed).
ConsiderÞrst the case of an evenm: m= 2p. The score ofO will be the highest ratio of points on either side of O. Obviously if there are k points on one side (and m−k on the other side), 0≤k≤p, this number is(m−k)/m, and this happens with probability 2 1
2m à m
k
!
when 0≤k ≤p−1, and with probability 1 2m
à m k
!
when k =p. Hence
Eρ(O) = 1 2m
à 2p p
! p
2p + 1 2m−1
p−1
X
k=0
à m k
!m−k m
= 1
2m+1 Ã 2p
p
! + 1
2m−1
p−1
X
k=0
à m−1 k
!
and the second term on the right-hand side is 1/2. Now if m= 2p+ 1:
Eρ(O) = 1 2m−1
Xp k=0
à m k
!m−k
m = 1
2m−1 Xp
k=0
à m−1 k
!
= 1
2m−1
p−1
X
k=0
à m−1 k
! + 1
2m
à m−1 p
!
| {z }
=12
+ 1 2m
à m−1 p
!
All we used in this proof was that the original distribution is symmetric about O and that some degeneracies occur with probability zero. Hence the result, which is
distribution-free. ¥
Bidimensional campaigns
We consider the same process as in Proposition 1 and choose m random points in a disk centered at O: Q1, Q2, . . ., Qm. For each i, 1 ≤ i ≤ m, we set Pi equal to Qi or to
−Qi with equal probability 1/2 (without loss of generality, we can choose theQi’s on the same side of a hyperplane through 0 as in Figure 4 below; Figure 5 corresponds to the conÞguration: Pi = Qi for i = 1,3,4 and Pi = −Qi for i = 2,5). The points P1, . . ., Pm are again i.i.d. random points in the disk. The original question answered by Wendel (1962), see also Wagner and Welzl (2001), was: what is the probability that O is not in the convex hull of the Pi’s? (In other words, what is the probability that the score of O be 1?) The answer is: m/2m−1. Indeed, independently of the choice of theQi’s (again, we ignore the degenerate conÞgurations, occurring with probability zero, where two vectors Qi and Qj would be collinear), there are 2m possibilities to choose the signs of the Pi’s such that O can be separated from these points by a line (every partition of the Qi’s by a line through Ogives two such possibilities). Again, all we used in this line of reasoning was that the original distribution is symmetric aboutOand that some degeneracies occur with probability zero. Hence the result is, once more, distribution-free.
Figure 4 Figure 5
•
•
• •
•
•
•
•
• •
• Q1 G
Q2
Q3
Q4
Q5
−Q1
−Q2
−Q3
−Q4
−Q5
•
•
•
•
•
•
P1 G
P3
P4
P2
P5
We now proceed along this line of reasoning and compute the probability that the score of O be j/m. E.g., in the social choice conÞguration shown on Figure 5, the score of O is 4/5 as shown by the dotted separation line.
Each social choice conÞguration can be described as a (q, p)-sequence (or random
walk) of plus ones and minus ones, according to whether Pi is equal to Qi (then +) or not (then -), 1≤i ≤ m: M1, . . . , Mm, with, say, q plus ones andp minus ones, q+p=m.
The conÞguration on Figure 5 corresponds to the (3,2)-sequence: +−+ +−. The partial sumsk =M1+. . .+Mk represents the difference between the number of pluses and minuses occurring at the Þrst k places, 0 ≤ k ≤ m, with s0 = 0 and sm = q − p. DeÞne:
¯
s = maxksk ands = minksk.
Lemma 1 In the social choice conÞguration represented by a(q, p)-sequence (M1, . . . , Mm), the score of O is ρ(O) = max{q−s, p+ ¯s}
m .
Proof of Lemma 1: Consider a line which separates the+Qi’s from the−Qi’s and passing through O. It has q of the Pi’s on one side and p on the other side. Now turn this line by pivoting at O so that it goes in-between Q1 and Q2: it has now q−s1 of the Pi’s on one side and p+s1 on the other side. Now turn it by pivoting at O so that it goes in-between Q2 andQ3: it has now q−s2 of the Pi’s on one side andp+s2 on the other side. And so on. The maximum number ofPi’s on one side of a line throughOis therefore max{. . . , q−sk, . . . , p+sk, . . .}= max{q−s, p+ ¯s}. Hence the result. 2
To compute the probability that the score of O be j/m, we need to compute the number of(q, p)-sequences such thatmax{q−s, p+¯s}=j. To do that, we follow a classical geometric method in the standard orthonormal basis where the x-axis in horizontal and the y-axis is vertical. Following Feller (1968), the sequence (M1, . . . , Mm) is identiÞed with apathfrom the origin to the point(m, q−p): this path is a polygonal line whose vertices have abscissa0,1, . . . , mand ordinatess0, s1, . . . , sm =q−p; ¯sis the highest point of the path and s the lowest. Obviously there are9
à m p
!
such paths from the origin to the point (m, q−p): as many as there are ways of choosing the p places for the minuses out of them possibilities.
Note that for a (q, p)-sequence, s¯≥max{0, q−p} and s ≤min{0, q−p} entail that max{q−s, p+ ¯s} ≥ max{q, p}, therefore we restrict attention to j ≥ max{q, p}. And in the case when m is even and p = q = m/2, obviously max{q−s, p+ ¯s} ≥ m/2 + 1 therefore we restrict attention in that case toj ≥m/2 + 1. Hence we considerj such that [m/2] + 1≤j ≤m.
A (q, p)-sequence is such that max{q −s, p+ ¯s} = j whenever the associated path remains in the corridor between the lines y =j −p and y= q−j, and hits at least one of them (see Figure 6 drawn for the conÞguration of Figure 5 andj = 4).
9By convention, the combination number will be set to zero in casep <0or p > m.
Figure 6
•
•
•
•
•
•
(m, q−p)[= (5,1)]
y=j−p[= 2]
y=q−j[=−1]
O -
Lemma 2 Fix j, [m/2] + 1 ≤ j ≤ m. The number of (q, p)-paths such that max{q − s, p+ ¯s}=j is
am,q,j =
( Am,q,j+1−Am,q,j if m−j ≤q ≤j
0 otherwise
where for m−j ≤q ≤j Am,q,j =X
k
"Ã
m q+k(2j−m)
!
−
à m
j +k(2j−m)
!#
(1) (the series extending over all integers k from −∞ to +∞, but having only Þnitely many non-zero terms) is the number of (q, p)-paths such that max{q −s, p + ¯s} < j: those which hit neither y = j −p nor y = q− j, and the number of (q, p) paths such that max{q−s, p+ ¯s}< m+ 1 is given by
Am,q,m+1 = Ã m
q
!
. (2)
Proof of Lemma 2: The equation relating the am,q,j’s to the Am,q,j’s is immediate. The computation of theAm,q,j’s relies on repeated applications of the ‘reßection principle’ due to Désiré André (see, e.g., Feller (1968), Chapter III) and is done in the appendix. ¥ Proposition 2 Consider an m-sample independently drawn from distribution which is symmetric about O, then the probability that the score of O be expected value of the score of O be j/m, [m/2] + 1≤j ≤m is
¯
am,j = 1 2m
Xj q=m−j
am,q,j
and the expected value of the score of O is Eρ(O) =
Xm j=[m/2]+1
j¯am,j
m .
Proof of Proposition 2: Immediately follows from Lemma 2. ¥
0 5 10 15 20 25 30 35 40 45 50 55
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Expected min-max rate upper bound for d=1 and d=2
Expected min-max upper bound
m: number of voters
d=1 d=2
Figure 7: The min-max rate forn→ ∞ when d= 1 andd = 2. ForÞniten, these Þgures are upper bounds for the actual min-max rates.
5 Concluding Comments
The present paper proposes a theorem of aggregation of individual preferences through the 50% majority rule in a multidimensional spatial voting model. Of course, the result is obtained at a non-negligible cost in terms of assumptions: Þrst the regularity of the simplicial distribution of voters’ ideal points; second the ‘uniform distribution’ on the set of linkages between issues imputed by ideologies (the Haar probability measure on G(n, d)). Of course, the robustness of the results when one relaxes these two assumptions should be studied. But on the other hand it is quite strong since it gives existence for
the 50% majority rule. Another important aspect is that it Þngers the mean voter as the candidate most likely to be stable in the voting process. As underlined in Crès and Tvede (2005), mean voter theorems are very welcome in public economics because in many contexts the mean voter is the one who has the right incentives as far as making an economically efficient choice is concerned.
We would like to stress one last point. An important property of the approach chosen here is that it is compatible with the idea that politicians “die in their ideological boots”.
Poole (2003) shows a variety of evidence that members of the US Congress are ideologically consistent: they adopt an ideological position and maintain it over time. An interpretation of Theorem 2 is that in the long run, ignoring the historical, sociological and mediatic shocks which are going to shape the linkages between political issues, this might be a good strategy not to change one’s mind! A strategic politician should choose an ideological position that he/she believes will place, as frequently as possible over the years, his/her imputed platform at the center of gravity of the voters’ ideal points. Different ideological positions come from different tastes, but also from different priors on the distribution of historical shocks (the so-called ‘sens de l’histoire’) and therefore of linkages between issues. Maintaining that ideological position over time is essential for their credibility, and thus an important asset for future political successes. Now, what is a good strategy in the short-run? A strategy here is neither the choice of an ideological position (basically chosen once for all at the beginning of one’s career –although there might be more than one beginning...) nor the choice of a political platform (automatically imputed by the ideological position), but an action that impacts the linkages between issues in a way that places the candidate at the center of gravity of the projected set of voters’ ideal points.
This is left for future work.
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Appendix
Proof of Lemma 2: The numbers Am,q,j of (q, p)-paths such that max{q−s, p+ ¯s} < j remain to be computed. These computations are based on the ‘reßection principle’ due to Désiré André (see, e.g., Feller (1968), Chapter III). Let A = (α, a) and B = (β, b) be points in the positive orthant: β > α ≥0, a >0, b > 0. By reßection of A on the x-axis is meant the point A0 = (α,−a) (see Figure 8).
Reßection principle: The number of paths from A to B which touch or cross the x-axis equals the number of all paths fromA0 toB.
Let N(m, c) =
à m
m−c 2
!
= N(m,−c) denote the number of paths from O = (0,0) to (m, c). Let a and b be positive, and −b < c < a. By the reßection principle, the number of paths from (0,0) to (m, c) which touch or cross y =a is equal to the number of paths from (0,2a)(the reßection of O on the axis y =a) to(m, c), i.e., N(m,2a−c).
By the same argument, the number from (0,0)to (m, c) which touch or cross y =−b is N(m, c+ 2b) =N(m,2a−c−2(a+b)).
Figure 8
•
•
•
• A
B
T A0
-
Now, by a double application of the reßection principle, a path from (0,0) to (m, c) which touch or crossy =aand theny =−b(called an ‘(ab)’ path in the sequel) can beÞrst associated to a path from (0,2a)to (m, c), itself associated to a path from(0,−2a−2b) to (m, c); hence N(m, c+ 2(a+b))of ‘ab’ paths. A triple application allows through the same line of argument to denumerate the paths which touch or crossy =a, theny =−b, then y =a again (‘(ab)a’ paths); their number is N(m,2a−c+ 2(a+b)). An extension of this method gives:
• N(m, c+ 2k(a+b)) for the number of paths which touch or crossy = a and then y=−b k times in a row (‘k(ab)’ paths);
• N(m,2a−c+ 2k(a+b)) for the number of paths which touch or cross y = a and then y=−b k times in a row and then y=a again (‘k(ab)a’ paths);
• N(m, c−2k(a+b)) for the number of paths which touch or crossy =−band then y=a k times in a row (‘k(ba)’ paths);
• N(m,2a−c−2(k+ 1)(a+b))for the number of paths which touch or crossy=−b and theny=a k times in a row and then y=−b again (‘k(ba)b’ paths).
Our aim is to compute the number of paths from(0,0)to(m, c)which touch or cross neither y = a nor y = −b. This comes Þrst by exclusion of paths which touch or cross y=aand paths which touch or crossy=−b. But thus ‘(ab)’ and‘(ba)’ paths are excluded twice and must be reincluded once. But then ‘(ab)a’ and‘(ba)b’ are excluded twice, then reincluded twice, and therefore must be re-excluded once... This standard application of
theinclusion-exclusion principle (see Comtet (1974), Chapter IV), leads to the formula:
N(m, c) − N(m,2a−c) +X
k>0
[N(m, c+ 2k(a+b))−N(m,2a−c+ 2k(a+b))]
− X
k>0
[N(m, c−2k(a+b))−N(m,2a−c−2k(a+b))]
for the concerned number, which can be rewritten:
X
k
[N(m, c+ 2k(a+b))−N(m,2a−c+ 2k(a+b))]
(over all integerskfrom−∞to+∞, but onlyÞnitely many non-zero terms). The formula 1 obtains readily by substitution of the right parameters. ¥
